The edge index \(n(\mathcal {C}_R, \mu )\)  can be interpreted as quantized pumping at the edge when both \(k_x\)  and \(a\)  are driven along the loop \(\mathcal {C}_R\) , although it is not clear how to implement such a loop in practice. Other loops of \(\mathring{C}\) , in particular at fixed momentum \(k_x\) , also lead to a quantized value. See Section  for more details.
The proof of Thm. REF  is done in Section . We use the same approach as in [1]}, based on scattering amplitudes and Levinson's theorem, originally developed in [2]} for discrete models.
